Zeroes of Sine and Cosine
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Theorem
- $(1): \quad \forall n \in \Z: x = \left({n + \dfrac 1 2}\right) \pi \implies \cos x = 0$
- $(2): \quad \forall n \in \Z: x = n \pi \implies \sin x = 0$
Proof
From Sine and Cosine are Periodic on Reals: Corollary:
$\cos x$ is:
- strictly positive on the interval $\displaystyle \left({-\frac \pi 2 \,.\,.\, \frac \pi 2}\right)$
and:
- strictly negative on the interval $\displaystyle \left({\frac \pi 2 \,.\,.\, \frac {3 \pi} 2}\right)$
$\sin x$ is:
- strictly positive on the interval $\left({0 \,.\,.\, \pi}\right)$
and:
- strictly negative on the interval $\left({\pi \,.\,.\, 2 \pi}\right)$
The result follows directly from Sine and Cosine are Periodic on Reals.
$\blacksquare$
